Find values for which the limit exists

  1. Hi,
    I'm given this problem:

    Find conditions for variables a, b, c so that the limit

    \lim_{[x,y] \rightarrow [0,0]} \frac{xy}{ax^2 + bxy + cy^2}


    What I have only found so far is that for all variables non-zero the limit doesn't exist. Anyway, I have no clue how to find the conditions for which it does. I tried a = b = c = 0, but it doesn't seem to help to me...

    Thank you for the enlightenment.
  2. jcsd
  3. Lisa!

    Lisa! 882
    Gold Member

    OOps I was too late in deleting my post. Actually I made a mistake in solving that and yes, what makes me unsure is that I don't think we could use hopital rule for these kind of limit.
  4. Lisa!

    Lisa! 882
    Gold Member

    You'd better to ask another homework helper, but I solve it in another way now . If you look at numerator nad denominator, you see both of them have xy. So?
  5. Ok, now it seems to me that the condition for the limit to exist is that a = c = 0.

    Anyway, it is just the result of guessing method, is there any more exact approach to solve this?
  6. HallsofIvy

    HallsofIvy 41,255
    Staff Emeritus
    Science Advisor

    For functions like this, where you have two variables, I find it best to convert to polar coordinates. That way, exactly one variable, r, measures the distance to (0,0) which is the crucial factor. In polar coordinates,
    [itex]x= r cos(\theta)[/itex] and [itex]y= r sin(\theta)[/itex] so
    [itex]xy= r^2 cos(\theta)sin(\theta)[/itex], [tex]x2= r^2 cos^2(\theta)[/itex], and [tex]y^2= r^2 sin^2(\theta)[/itex].
    Of course, then [itex]ax^2+ bxy+ cy^2= ar^2cos^2(\theta)+ br^2sin(
    theta)cos(\theta)+ cr^2sin^2(\theta)[/itex] so that
    [itex]ax^2+ bxy+ cy^2= r^2(acos^2(\theta)+ bsin(
    theta)cos(\theta)+ csin^2(\theta)[/itex].

    That means that
    [tex]\frac{xy}{ax^2+ bxy+ cy^2}= \frac{sin(\theta)cos(\theta)}{acos^2(\theta)+ bsin(\theta)cos(\theta)+ csin^2(\theta)}[/tex].

    Notice that there is no "r" in that! This can have a limit as r-> 0 only if it does NOT depend on [itex]\theta[/itex]- it is a constant. One obvious choice for a,b,c is a= c= 0, b= 1 but there may be others.
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